A New Moment Method for Solving the Coagulation Equation for Particles in Brownian Motion

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1 Aerosol Science and Technology ISSN: (Prin (Online Journal homepage: A New Moment Method for Solving the Coagulation Equation for Particles in Brownian Motion Mingzhou Yu, Jianzhong Lin & Tatleung Chan To cite this article: Mingzhou Yu, Jianzhong Lin & Tatleung Chan (8 A New Moment Method for Solving the Coagulation Equation for Particles in Brownian Motion, Aerosol Science and Technology, 4:9, 75-73, DOI:.8/ To link to this article: Published online: Aug 8. Submit your article to this journal Article views: View related articles Citing articles: 83 View citing articles Full Terms & Conditions of access and use can be found at

2 Aerosol Science and Technology, 4:75 73, 8 Copyright c American Association for Aerosol Research ISSN: print / online DOI:.8/ A New Moment Method for Solving the Coagulation Equation for Particles in Brownian Motion Mingzhou Yu,, Jianzhong Lin,, and Tatleung Chan 3 The State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou, China China Jiliang University, Hangzhou, China 3 Department of Mechanical Engineering, the Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong A new numerical approach for solving coagulation equation, model, is first presented. In this model, the closure of the moment equations is approached using the Taylor-series expansion technique. Through constructing a system of three first-order ordinary differential equations, the most important indexes for describing aerosol dynamics, including particle number density, particle mass and geometric standard deviation, are easily obtained. This approach has no prior requirement for particle size spectrum, and the limitation existing in the log-normal distribution theory automatically disappears. This new approach is tested by comparing it with known accurate solutions both in the free molecular and the continuum regime. The results show that this new approach can be used to solve the particle general dynamic equation undergoing Brownian coagulation with sufficient accuracy, while less computational cost is needed.. INTRODUCTION In theory, the evolution of aerosol size distribution in the flow field can be highly traced through solving the particle general dynamic equation (PGDE. This PGDE equation has an ability to describe the combined physical processes for very fine particles, such as the internal processes of nucleation, condensation, growth, and coagulation as well as the external transport processes of diffusion, convection, and thermophoresis. Unfortunately, the general dynamic equation is none other than Boltzmann s transport equation, in which only a limited number of known analytical solutions exist due to its own non-linear integro-differential structure. Hence, an alternative method, the numerical technique, has to be used to obtain approximate solution for this problem. However, the numerical calculations often become impractical, even with a modern super-computer, due to Received November 7; accepted 7 May 8. We would like to acknowledge the financial supported by the Major Program of the National Natural Science Foundation of China (Grant No Address correspondence to Jianzhong Lin, China Jiliang University, Hangzhou, 38, China. mecjzlin@public.zju.edu.cn the requirement of large computational cost. In order to break the limitation in computational cost, three prominent methods were used, i.e., the moment method (MM (Pratsinis 988; Hulbert and Katz 994, Chan et al. 6, Lin et al. 7, the sectional method (SM (Gelbard and Seinfeld 98; Talukdar and Swihart 4, and the stochastic particle method (SPM (Wells and Kraft 5; Morgan et al. 6. These methods have both advantages and disadvantages in accuracy and efficiency, and now they are used in different fields in terms of the particular requirement. Because of the relative simplicity of implementation and low computational cost, the MM has been extensively used to solve most particulate problems, and has become a powerful tool for investigating aerosol microphysical processes in most cases (Wright et al. ; Yu et al. 8b. As a key issue for the MM, the closure of the moment evolution equations has been achieved by making a prior assumption for the shape of the aerosol size distribution (Pratsinis 988, approximating the integral moment by an n-point Gaussian quadrature (McGraw 997 and assuming the pth-order polynomial form for the moments (Barrett and Jheeta 996. Typically, the moment method proposed by Pratsinis ( 988 and the quadrature method of moments (QMOM proposed by McGraw (997 are two widely used moment methods now. The is the early form of the MM, and has been developed to obtain the representative properties of the aerosol size distribution such as total particle concentration, mean size, and polydispersity only through solving the first three moments. However, the introduction of prior assumption with log-normal distribution in the will lead to the method invalid at early stage of particle formation (Yu et al. 8a, 8b. The QMOM, to some extent, is an improved form of the and has been proved to be applicable to any form of the growth laws and coagulation kernels (Upadhyay and Ezekoye 3. When the population balance is written in terms of one internal coordinate (e.g., particle length or size, the closure problem can be successfully solved with the use of a quadrature approximation without prior assumption. However, 75

3 76 M. YU ET AL. the weights and abscissas of the quadrature approximation must be additionally obtained using the product-difference algorithm (Gordon 968, which greatly increases the computational cost in contrast to the, especially in the case where the PGDE is required to be incorporated into the computational fluid dynamics (CFD codes. In recent years, the particle formation occurring in a chemically reacting flow has been increasingly received attention, and accordingly it is possible to simultaneously capture the details of the fluid flow and transport, the evolution of the particle size distribution and complex chemical kinetics (Yu et al. 8a, 8b. For these particulate systems, consuming large computational cost will be inevitable for further investigations. Although some researchers such as Friedlander ( and Frenklach ( have shown the moment method without requiring a priori knowledge of the size spectrum is not new, it is still necessary to construct a new approach with respect to moment equation, which is easy to implement with low computational cost like the and has not the prior requirement for particle size spectrum like the QMOM, to adapt to the requirement of modern complicated particulate industries. In this article we present a new approach to solve the PGDE undergoing Brownian coagulation. The underlying idea of the approach is that the closure of the moment evolution equations is approached using the Taylor-series expansion technique. The Taylor-series expansion method of moments ( has no prior requirement for particle size spectrum, and the number of moment equations needed is equal to the order of the Taylorseries expansion. Theoretically, the precision of solution and the computational cost increase with increasing the terms of Taylorseries expansion. Based on the trial numerical simulation, keeping 3 terms of Taylor-series is preferable if both precision and efficiency are simultaneously considered. The present investigation is concerned with the Brownian coagulation both in the free molecule and the continuum regime. Additionally, it should be pointed out that the is easy to be extended to investigate other aerosol dynamics such as coalescence, surface growth and nucleation, and the derivation of Bivariate extension of the simultaneously tracking the particle size and surface is also easy to be obtained, which is similar to the QMOM method (Wright et al... THEORY The scientific description for ultrafine particle dynamics starts from Schmoluchowski s discrete coagulation equation (Smoluchowski 97 and follows the integro-differential equation governing the continuous size distribution for the number concentration function developed by Muller (98. It is believed that the Muller s equation remains the absolute predominated status for investigating aerosol Brownian coagulation problems since its appearance in 98, which is written as: n(v, t = v β(v, v v n(v, n(v v, dv n(v, β(v, vn(v, dv, [] where n(v,dv is the number of particles whose volume is between v and v+dv at time t, and β(v, v isthe collision kernel for two particles of volumes v and v.inprinciple, depending on the functional form of the coagulation kernel and on the method of solution of the Smoluchowski equation, there can be various analytical solutions for Equation (. The general disposition for this problem is to transform Equation ( into an ordinary differential Equation with respect to the moment m k. The moment transformation involves multiplying Equation ( by v k and then integrating over the entire size distribution, and finally the transformed moment Equation based on the size distribution are obtained (Upadhyay and Ezekoye 3: m k t = [ (v + v k v k v k ] β(v, v n(v, n(v, dvdv, (k =,,, [] where the moment m k is defined by: m k = v k n(vdv. [3] In the past, some efforts have been made to achieve the closure of Equation (. Three prominent methods, i.e., making a prior assumption for the shape of the aerosol size distribution (Pratsinis 988; Yu et al. 6, approximating the integral moment by an n-point Gaussian quadrature (McGraw 997; Yu et al. 7, and assuming the pth-order polynomial form for the moments (Barrett and Jheeta 996, have been evaluated and compared (Barrett and Webb 998. In this article, we develop a new approach, Taylor expansion technique, to solve this problem... Brownian Coagulation in the Free Molecule Regime In this size regime the collision frequency function β FM is (Pratsinis 988: β FM = B (/v + /v /( v /3 + v /3, [4] where B = (3/4π /6 (6k b T/ρ /, k b is the Boltzmann constant, T is the gas temperature and ρ is the mass density of the particles. Let us first concentrate on (/v + /v / in Equation (4 and then rewrite it in the form: C = (/v + /v /. [5] In principle, as Equation (4 is introduced into Equation (, at the moment Equation ( cannot yet be closed because of the presence of the quadratic form of the variable C. Usually, Equation ( is brought to an integrable form using the approximation (Pratsinis and Kim 989; Lee and Chen 984: C = b ( /v / + /v /, [6]

4 ANEW MOMENT METHOD FOR BROWNIAN COAGULATION 77 where b is a strong function of the spread of the aerosol size spectrum. In this study, we expand Equation (5 in Taylor-series, then substitute it into Equation (, which is subsequently solved. In general, it is more convenient to expand (v + v / in binary Taylor-series than (/v + /v /. So, we are more willing to rewrite Equation (4 in the following particular form: β FM = B (v + v /( v /6 v / + v /6 v /6 + v / v /6. [7] Here, there is only necessary to expand (v + v / for Equation (7 in a Taylor-series about point (v = u, v = u. Using the l Hôpital s rule, it can be verified that the Taylor-series expansion converges in the interval (, u. In this study, we define the expansion point u as the mean particle size v. Atthe case, the convergence of the Smoluchowski Equation is highly dependent on a number of factors, including the size range of the particles, and on the time differencing scheme employed in the numerical simulation. In this study, we use fully implicit time differencing scheme to solve this stiff system. Under the condition of Taylorseries expansion, (v + v / should be: (v + v / = (v u u + 4 (v u + u 4 u (v u (v u(v u 3u 3/ 6u 3/ (v u + [8] 3u 3/ For Equation (8, we take the first three terms of Taylor-series expansion, which is consistent with the following treatments for the fractional moment. Disposing Equation (7 with Equation (8 and substituting it into Equation ( results in the following complicated equation with respect to the first three moments m, m, and m : where dm = B [ξ φ + ξ φ + ξ 3 φ 3 ]n(v, n(v, dvdv [9.] dm = [9.] dm = B [ς φ + ς φ + ς 3 φ 3 ]n(v, n(v, dvdv [9.3] ξ = v/6 v + v /6 v /6 ξ = v7/6 v + v5/6 v /6 + v/6, v + v5/6 v7/6 + /6 v + vv /6 v + v v /6, ς 3 = 4v 5/6 v 5/6 + v 7/6 3/ v + v v /6 + v 3/ v /6 + v v 7/6 + v3/6 + v/6 v v /6 + v3/6 + v/6 v v, /6 ς = 4v 5/6 v 5/6 + v 7/6 v + v 7/6 v, ς = v v 3/6 + 4v 5/6 v /6 + 4v 5/6 v /6 + v 3/6 7/6 v + v v 3/ + v 7/6 v 3/, ς 3 = 8v /6 v /6 + 4v 3/6 v 3/ + v 5/ v 7/6 + v 5/ v 7/6 + 4v 3/6 v 3/ + v v 9/6 + 4v 5/6 v 7/6, + vv 9/6 + 4v 5/6 v 7/6 φ = 3 u 8,φ = 3 8 u and φ 3 = 3u 3/. Using the transformed Equation (3, the right terms of Equation (9 can be integrated. At the case, their detailed expressions take the following form: where dm = B [ξ φ + ξ φ + ξ3 φ 3] [.] dm = [.] dm = B [ς φ + ς φ + ς3 φ 3] [.3] ξ = m /6m / + m /6 m /6 + m /6 m /, ξ = m 7/6m / + m 5/6 m /6 + m 5/6 m /6 + m 7/6 m / + m / m /6 + m / m /6, ξ = 4m 5/6m 5/6 + m 7/6 m / + m 3/ m /6 + m 3/ m /6 + m / m 7/6 + m 3/6 m /, +m /6 m /6 + m 3/6 m / + m /6 m /6, ς = 4m 5/6m 5/6 + m 7/6 m / + m 7/6 m /, ς = m /6m 3/6 + 4m 5/6 m /6 + 4m 5/6 m /6 + m 3/6 m / + m 7/6 m 3/ + m 7/6 m 3/, ς3 = 8m /6m /6 + 4m 3/6 m 3/ + m 5/ m 7/6 + m 5/ m 7/6 + 4m 3/6 m 3/ + m / m 9/6 + 4m 5/6 m 7/6 + m / m 9/6 + 4m 5/6 m 7/6 For all the coefficients from ξ to ς 3,itisclear that all the same terms are not grouped and all the terms completely correspond to the coefficients from ξ to ς 3 in Equation (9. In particular, Equation (9 denoted by these coefficients is absolutely composed with fractional-order moments, not integral moments. In order to further close the moment equation, it is necessary to dispose these fractional moments. Therefore, we continue to use the Taylor-series expansion technique to dispose Equation (3 with respect to fractional kth moment. In the Equation (3, v k can be expanded with Taylor-series about point v = u (this is

5 78 M. YU ET AL. consistent with the expansion point u in (v + v / : v k = u k +u k k(v u+ uk k(k (v u +. [] Similar to the disposition for (v + v /,wetake the first three terms of Taylor-series. In order to meet the requirement of moment transformation, it is necessary to group the terms of truncated Equation (: ( u v k k k = + uk k uk k v + ( u k k + u k kv + u k 3uk k. [] Substituting Equation ( into Equation (3, we have ( u m k = v k k k n(vdv = uk k + u k km + (u k + uk k 3uk k m + ( u k k m. [3] It is obvious that m k in Equation (3 can be considered as a function of the first three moments m, m, and m.inprinciple, the number of moments in Equation (3 is consistent with the reserved terms of Taylor-series in Equation (. For example, there is a need to take the first four terms of Taylor-series if m k is required to be a function of four moments m, m, m, and m 3.As the Equation (3 is substituted into Equation (, the moment equation merely comprised by integral moment variables can be easily obtained: dm = ( B 7m m /6 4m m m5/6 6859m 4 m /6 59m /6 [5.3] It is obvious that Equation (5 is a system of first-order ordinary differential equations and all the right terms are denoted by the first three moments m, m, and m, and thus this system can be automatically closed. Under these conditions, the first three moments, which are also the three predominate parameters for describing aerosol dynamics, are obtained through solving this first-order ordinary differential system. Here, it should be pointed out that the whole derivation of the equations does not involve any assumptions for particle size spectrum, while the final mathematical form is much simpler than the model... Brownian Coagulation in the Continuum Regime The disposition for aerosol Brownian coagulation equation in the free molecule regime can be also expanded to be in the continuum regime. In this regime the collision frequency function is (Barrett and Webb 998: ( β C = B v + (v /3 /3 v /3 + v /3, [6] where B = k bt, µ is the gas viscosity. Similar to the solution for Brownian coagulation in the free molecule regime, we µ substitute Equation (6 into Equation ( and have: dm dm dm = =, = B ( 4m m u 4388m u + 44m m u + 669m u4 65m + 69m m u 3 584u 3/6, [4.] [4.] B ( 6748m u + 7m 34m m u 76m m u 3 376m m u + 65m u4 59u /6 [4.3] Here, the expansion point of Taylor-series expansion is denoted by the mean particle size v(= m /m. At the case, Equation (4 can be written in the following form: dm = dm ( B 65m m 3/6 m m m7/6 93m 4 m/6 584m 3/6, [5.] =, [5.] dm dm dm = B (v /3 v /3 + + v /3 v /3 n(v, n(v, dvdv [7.] =, [7.] = B + v /3 v 4/3 ( v 4/3 v /3 + 4vv n(v, n(v, dvdv, [7.3]

6 ANEW MOMENT METHOD FOR BROWNIAN COAGULATION 79 and so on. After performing the moment transformation with Equation (3 together with the 3rd-order Taylor-series expansion for fractional moment, we can obtain the final form for aerosol Brownian coagulation in the continuum regime: dm = B ( 5m 4 + m m 3m m m m 8m 4, [8.] dm =, [8.] dm = B ( m m 3m m m 5m 4 8m [8.3] 3. COMPUTATIONS The numerical computations are all performed on an Intel (R Pentium 4 CPU 3. GHz computer with memory GB. The 4th-order Runge-Kutta method with fixed time step is used to solve the system of ordinary differential equations, and the Jacobi method is applied to obtain the eigenvalues and the eigenvectors of Jacobi matrix, which occurs in the computation of the QMOM. In the computations, all the numerical simulations are based on dimensionless time, and the time step is supposed to be small enough,., in order to allow an accurate result. The detailed dimensionless time is defined by τ = B t in the free molecular regime and τ = B t in the continuum regime. All the programs are written using the C Programming language and are performed on Microsoft Visual C++ 6. compiler. 4. RESULTS AND DISCUSSION We now test our new approach by comparing it with known accurate models (Barrett and Jheeta 996; Barrett and Webb 998 both in the free molecular and the continuum regime. In order to make better comparison, we have written the codes based on the QMOM model, the model and the model. For QMOM, five different cases are included, i.e., quadrature approximation with, 3, 4, 5, and 6 nodes. The used here is based on the log-normal distribution approximation developed by Pratsinis (988. The initial distribution of all cases is assumed to be log-normal, i.e., n(v, = N exp{ (ln v ln v g /w g }/{ πvw g } with N =, v g = 3/ and wg = ln(4/3, which is consistent with the initial distribution conducted by Barrett and Jheeta (996, Barrett and Webb (998, and Williams (986. Under the case with log-normal initial distribution, the initial kth moments m k are obtained through solving m k = Nv k g ek w g /. The calculation of relative error for any variables follows the definition: %Error = φ φ reference φ reference %, [9] where φ is the arbitrary variable and φ reference is the referenced variable. 4.. Brownian Coagulation in the Free Molecular Regime In this regime the computation time is scaled by the definition τ = B t, which is consistent with the disposition by Barrett and Webb (998. Here, we follow the study of Barrett and Webb (998 in which they compared various models at several special time points using tabulation approach. Table shows the values of zero moment m and second moment m given by various methods (the first moment m is a constant when only Brownian coagulation is considered at time τ =., 5., and.. In this table, the values are all from Table 3ofBarrett and Webb (998 except for the and the. For QMOM,, and models, it should be noted that our own codes give the same values with those provided by Barrett and Webb (998. The last column in Table shows the consumed CPU time up to τ =. Since some studies have shown that the QMOM model with or 3 nodes produces the accurate result, and the precision can be increased with increasing the number of nodes (McGraw 997; Upadhyay and Ezekoye 3, it is feasible to take the as a reference to TABLE Moments m and m in the free molecular regime at different dimensionless time and the consumed dimensionless time for various methods τ = τ = 5 τ = CPU time (B Method m m m m m m (τ = Laguerre quad QMOM points QMOM 3 points QMOM 6 points (Pratsinis Gamma (Williams (presen

7 7 M. YU ET AL.. m QMOM Error% 5 QMOM τ(b (a τ(b (b FIG.. The zero moment m and the relative error for various methods in the free molecular regime. The relative error denotes the ratio of m from various methods to the QMOM with 6 points. validate other models. In fact, all the approximate methods in Table give reasonable agreement with the reference results. For the zero moment, the results obtained by are all identical to those by the except at τ =.. At τ =., the result of is closer to than Gamma model, QMOM, and models. Although different models provide different results for the second moment, the model produces the same precision as and model. In addition, it can be easily seen that the consumed CPU time for the is the shortest among all the investigated models. In order to further validate this new method, it is necessary to have the comparisons between our model and other known models over large evolution time. The evolutions of m with scaled time τ together with the relative error in various methods are shown in Figure. The relative error denotes the ratio of m from the various methods to the. In fact, all the errors mentioned in this paper, including the zero and second moment, are all the relative values of,, or QMOM- to. In Figure a, all the curves overlap each other, and it is difficult to distinguish from one to the other. Hence, the differences between them must be evaluated using the relative error, Error%, which is shown in Figure b. From this figure, it can be found that the error for the is always less than % from τ = toτ =. In addition, it is clear that the curves of error exhibit the same trend for all the models, and more importantly, the gives the least relative error beyond τ = 6. For all the investigated models, the relative error reaches constant values at τ>6, indicating that these models can give the reasonable results for zero moment. The second moment m as well as the relative error are shown in Figure. Like the zero moment, the relative error is still denoted by the ratio of various methods to the. Similar to Figure a, all the curves overlap in Figure a. In Figure b, the curves of relative error are very similar for the and, except that values are negative for the former but positive for the latter. For the, the error initially increases in the negative direction and then holds nearly m 3 QMOM Error% QMOM τ(b (a τ(b (b FIG.. The second moment m and the relative error for various methods in the free molecular regime. The relative error denotes the ratio of m from various methods to the QMOM with 6 points.

8 ANEW MOMENT METHOD FOR BROWNIAN COAGULATION 7 σ g QMOM τ(b FIG. 3. The geometric standard deviation σ g in the free molecular regime for various methods. constant at.65%. For the, the error starts to show its largest value at τ =., and then decreases until it attains its nearly constant value at about τ =. However, the absolute values of relative error of the are less than that of the in the entire time range. This indicates the can achieve the same precision as the for solving the second moment, although the precision is little less than that produced by the. There exists an asymptotic solution for the distribution of reduced particle size, the so-called self-preserving size distribution (SPSD (Frenklach 985, which has become an important tool to explore the aerosol coagulation mechanisms. The selfpreserving form is usually approximated by a lognormal distribution with a geometric standard deviation σ g which is obtained by solving the following equation (Pratsinis 988: ln σ g = 9 ln ( m m m. [] Although Equation ( was originally derivated from lognormal theory, it may be also applied in QMOM and models. Figure 3 shows the variations of σ g with time in the free molecular regime for various methods. We can see that the SPSD attains for all methods at about τ = 5. For the QMOM, the asymptotic values of σ g decrease from.378 to.346 with an increase of nodes from to 6. The asymptotic value of.345 for the is the closest to the among all the investigated models. In addition, the asymptotic value of.355 for the is identical to the value given by Lee et al. (984 who used log-normal functions for particle size distribution. The above comparisons suggest that the has a higher precision in describing the asymptotic aerosol size distribution than other methods when compared to. 4.. Brownian Coagulation in the Continuum Regime Similar to the treatment for Brownian coagulation in the free molecular regime, we now compare various methods in the continuum regime. The results from the are considered to be exact and are used as references. The computational time in Equation (8 is scaled by τ = B t. The variations of the zero moment m and the relative error for various methods in the continuum regime are shown in Figure 4. In Figure 4a, all the curves overlap with each other and thus we cannot distinguish one from another. Hence, it is still necessary to use the relative error to distinguish the precisions produced by all the investigated models. In the whole range up to τ =, Figure 4b shows the model nearly exhibits the same curve with the model, and in particular, the results of are closer to the. Moreover, like the and QMOM, the relative error produced by converges to a nearly constant value beyond τ =, and more importantly, this error is always below.5%. The results clearly indicate that the is capable of handling the evolution of zero moment with high precision in the continuum regime. The variations of second moment and relative error for various methods are shown in Figure 5. Similarly, all the curves. 4 m QMOM τ(b (a Error% τ(b (b QMOM FIG. 4. The zero moment m and the relative error for various methods in the continuum regime. The relative error denotes the ratio of m from various methods to the QMOM with 6 points.

9 7 M. YU ET AL. 3. m QMOM τ(b (a Error% τ(b (b QMOM FIG. 5. The second moment m and the relative error for various methods in the continuum regime. The relative error denotes the ratio of m from various methods to the QMOM with 6 points. exhibit the same trend and there are no visible differences with each other in Figure 5a. In the entire evolution time up to τ =, it is clear that in all the investigated models, the QMOM model produces the largest error, but its value is always below.6%. These suggest all the investigated models can achieve very high precision for representing second moment in this regime. Although the error produced by the is slightly larger than the, their curves almost show the same trend. Like the zero moment, the error produced by the converges to a nearly constant value of.4% beyond τ =. Typically, the curves of other models show the same trend except that there are negative values below τ = 5 for the. From the above comparisons, for the second moment, it can be easily found that the gives relative low error close to the. In the continuum regime, the particle size distribution can also achieve the self-preserving size spectrum. All the studies (Lee et al. 997; Pratsinis 988; Lee 983; Lee et al. 984 showed that the asymptotic value based on the geometric standard deviation σ g is near to.3 if the log-normal distribution theory is used. In Figure 6, the values of the geometric standard σ g QMOM τ(b FIG. 6. The geometric standard deviation σ g in the continuum regime for various methods. deviation for various methods are presented and compared. It is obvious that all curves show the same trend and attain each asymptotic value at about τ =. However, like in the free molecular regime, the diversity for these asymptotic values still exists in this regime. For the QMOM, the asymptotic value of σ g decreases from.35 to.35 with an increase of nodes from to 6. In this figure, the curves for are not shown because their asymptotic values are always between QMOM and. Here, it is worth pointing out that the gives the same curve as the model, and their asymptotic values are the same value of.39, which is consistent with the asymptotic value of.3 reported in the literatures (Lee et al. 997; Lee 983; Lee et al This suggests that the has an ability to handle the evolution of particle size spectrum with time based on geometric standard deviation, in particular with the same precision as the model. 5. CONCLUSIONS In this study we used the Taylor-series expansion technique to dispose the collision terms and the fractional moments to obtain a new form for the moment equations. This approach requires no prior requirements for particle size spectrum, and the number of moment equations to be solved simultaneously is equal to the reserved terms of Taylor-series. The present study was focused on Brownian coagulation in the free molecular regime as well as in the continuum regime, and the new approach was validated by comparing it with known methods. In order to validate model in accuracy and efficiency, three important indexes of aerosol dynamics, i.e., the zero moment, the second moment and the geometric standard deviation were compared. The results show that the new approach can be used to solve particle general dynamic equation undergoing Brownian coagulation with sufficient accuracy, while less computational cost is needed.

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